Virtual Work

Virtual work, if mastered well, can make it very easy to solve certain problems. While normal solvingmethods can involve numerous calculations, virtual work sometimes enables you to solve a problem withjust one equation, and thats the benet. The downside is that its easy to make errors.1 Increasing the degree of freedomMost structures that occur in problems are statically determinant. They therefore can not move. To usevirtual work, you must increase the degree of freedom in such a way, that the structure can move. Buthow to do that is sometimes complicated.Every part in the structure can pass on forces. But which forces depends on the type of structure. Intable 1 is an overview of all the structure types, and which forces they can transfer. As you can see, thereare a lot of dierent connection types, some of which are familiar, some of which are not, but for whichyou just have to use your imagination to nd out what they look like. The important part is that theycan move in certain directions, and can not move in other directions.Structure type Normal forces Shear forces Bending momentsNormal bar X X XHorizontal sliding bar X XVertical sliding bar X XHinge X XHinge on wheel support X(tangential to the bar)Hinge on wheel support X(perpendicular to the bar)Sliding connection (unable to rotate) XNo connectionTable 1: Overview of structure types, and their degree of freedom.To use virtual work, you need to replace a certain connection by a connection that has just one moredegree of freedom (one less X). The additional degree of freedom given should be replaced by an externalforce or torque. Suppose you want to know the normal force caused by a normal hinge on wheel support,you have to replace it by just a force (no connection), acting in the same direction as the reaction forcecaused by the support was acting. Suppose you want to know the bending moment in a bar, you have toreplace part of the bar by a hinge and two moments (one on each side of the hinge, oppositely directed).2 Making the structure moveAfter the degree of freedom has been created, the structure should move. Just imagine that the structuremoves (rotates/slides) at the point at which youve replaced a connection, and draw it in the picture.Give the distance/angle that this point has moved a name (for example u for distances or for angles).Now draw the rest of the new structure, with the corresponding movements. But do keep in mindthat bars remain straight, xed hinges can not move, and hinges on wheel supports can only move in 1direction.Now look at every signicant point in the structure that has moved/rotated in any way. If it has moved,express the movement in u (if u hasnt been dened yet, just dene it as a certain movement). If ithas rotated, express the rotation in (dene it if necessary). Now you should have a drawing on whichis clearly visible what point moved what distance in what direction.13 Setting up the equationThe most important thing now, is to set up the virtual work equation. As you should know, work isforce times distance traveled (A = F u). Also, work is torque times angle rotated (A = M ) ( inradians!). It is very important to remember the following rule: If a force points in the same direction asthe movement (thus the force partially causes the movement), the work done is positive. If a force pointsin the opposite direction as the movement (thus the force partially counteracts the movement), the workdone is negative. And if the force is perpendicular to the movement, then the work done is 0. In formula,this can be written as A = F u cos , where is the angle between the force and the movement.4 Working out the equationWhen the work done by all the forces and the torques has been written down (with the right sign),it should be equal to 0, since the structure is in equilibrium. Now the equation should be solved.However, next to the unknown force/torque, there are often 2 unknown variables: u and . To solvethe equation, you rst have to express one of those variables in the other one. When doing this, the smallangle approximation should be used: tan = . This makes solving the equation a lot simpler.When one of the two unknown variables has been lled in, the equation should be solved. While solvingit, the other unknown variable will also disappear (if youve done everything right), so that the onlyunknown left is the unknown force/torque, expressed in other known forces and torques. Of course youneed to evaluate your answer (check whether it has the right unit, etc.), but if you have followed thesteps, the answer should be right.2